Eringen’s nonlocal elasticity theory is one of the most attractive approaches to investigate the intrinsic scale effect of nanoscopic structures. Eringen proposed both integral and differential nonlocal models which are equivalent to each other over unbounded continuous domains. Although the Eringen nonlocal models can be used as very useful tools for modeling the mechanical characteristics of nanoscopic structures, however, several researchers have reported some paradoxical results when they used the nonlocal differential model. In this paper, we develop a well-posed nonlocal differential model for finite domains, and its applicability to predict the static and dynamic behavior of a nanorod is investigated. It is shown that the proposed integral and differential nonlocal models are equivalent to each other over bounded continuous domains, and the corresponding elastic problems are well-posed and consistent. In addition, some paradigmatic static problems are solved the and we show that the paradoxical results disappear by using the present model.

中文翻译：

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有限域的适定非局部弹性模型及其在纳米棒力学行为中的应用

Eringen 的非局部弹性理论是研究纳米结构的内在尺度效应的最有吸引力的方法之一。Eringen 提出了在无界连续域上彼此等效的积分和微分非局部模型。尽管 Eringen 非局部模型可以用作模拟纳米结构力学特性的非常有用的工具，但是，一些研究人员在使用非局部微分模型时报告了一些自相矛盾的结果。在本文中，我们为有限域开发了一个适定的非局部微分模型，并研究了它在预测纳米棒静态和动态行为方面的适用性。结果表明，所提出的积分和微分非局部模型在有界连续域上彼此等效，并且相应的弹性问题是适定且一致的。此外，解决了一些典型的静态问题，我们表明使用本模型可以消除矛盾的结果。